62 Mr Brill, On certain Points specially [Feb. 24, 



tied above they are of the character described. Every locus of 

 the form x ±iy = const, will cause the expression for ds to vanish, 

 but it is only such of these loci that pass through the points 

 under discussion that form part of the locus of ultimate inter- 

 section of the members of either of the families. 



4. As the demonstration of the preceding article is attended 

 with some difficulties, it will be perhaps well to give another 

 demonstration modelled on one given by Kummer in the paper 

 referred to in the communication to which this is a sequel. If 

 we write 



£ + n = 2m and f 17 = v 2 , 



then we may consider the parameters of our two families of curves 

 as the roots of the equation 



a 2 - 2aw + v 2 = 0. 



To find the loci of ultimate intersections of the curves given by 

 this equation, we have 



a — u = ; 



and the loci we are in quest of are given by 



w a = v 2 , 



or u = + v. 



We will now discuss the direction of the tangent of one of 

 these loci. We have 



du du dy _ \dv dv dy\ 

 dec dy dx ~ \dx dy dx) ' 



du dv 

 dy _ dx~ dx 

 dx du dv ' 



dy-dy 



It remains to express the condition of orthogonality of the two 

 families of curves. We have the equations 



3| dt) _ du 



dx dx dx ' 



dx dx dx' 



and from these it follows that 



^-^dx =2 \Zdx- v dx)' ^-^ = - 2 r^~^r 



